Viewing mutual information as a utility measure, we may consider maximizing mutual information under local differential privacy as yet another privacy-utility trade-off. To formalize this, let \(X\sim P_X\) . The goal is to characterize the supremum of \(I(X; Z)\)  over \({\mathsf {K}}\in {\mathcal {Q}}_{\varepsilon , \delta }\) , i.e., the maximum information shared between \(X\)  and its \((\varepsilon , \delta )\) -LDP representation. Such mutual information bounds under local DP have appeared in the literature, e.g., McGregor et al. [1]} provided a result that roughly states \(I(X; Z)\le 3\varepsilon \)  for \({\mathsf {K}}\in {\mathcal {Q}}_{\varepsilon }\)  and Kairouz et al. [2]} showed for sufficiently small \(\varepsilon \)
\(\sup _{{\mathsf {K}}\in {\mathcal {Q}}_{\varepsilon }}I(X; Z)\le \frac{1}{2}P_X(A)(1-P_X(A))\varepsilon ^2,\) 
